
In order to control the R.H.S. of (REF ),
using the interpolation inequality (cf. [1]}) and (REF ),
\(\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(p)} \Vert _{L_t^1L^p_x}& \lesssim \Vert  |{\nabla }|^{2{\alpha }_1-1} d_{q+1}^{(p)} \Vert _{L_t^1L^p_x}\\& \lesssim \Vert d_{q+1}^{(p)}\Vert _{L_t^1L^p_x} ^{\frac{3-2{\alpha }_2}{2}} \Vert d_{q+1}^{(p)}\Vert _{L_t^1W^{2,p}_x} ^{\frac{2{\alpha }_2-1}{2}}\\& \lesssim \ell ^{-1}\lambda ^{2\alpha _2-1}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\tau ^{-\frac{1}{2}}.\) 
